A060887 -id:A060887 - cp's OEIS frontend

A020521 13th cyclotomic polynomial evaluated at powers of 2.

13, 8191, 22369621, 78536544841, 300239975158033, 1190112520884487201, 4797324681010433232961, 19495118728903626376364161, 79538861190790864407636279553, 325153619321163373997995856232961, 1330527338889299954891005307651097601

Offset: 0

Simon Plouffe

nonn

Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. Cites this sequence.

Cf. A000079, A060887.

with(numtheory,cyclotomic):seq(cyclotomic(13,2^i),i=0..24);

Table[Cyclotomic[13,2^n],{n,0,24}] (* Benedict W. J. Irwin, Jul 15 2016 *)

a(n) = polcyclo(13, 2^n); \\ Michel Marcus, Apr 06 2016

a(n) = A060887(A000079(n)). - Michel Marcus, Apr 06 2016

G.f.: Sum_{k=0..12} 1/(1-2^k*x). - Benedict W. J. Irwin, Jul 15 2016

A342689 Square array read by antidiagonals (upwards): A(n,k) = (k^Fibonacci(n) - 1) / (k - 1) for k >= 0 and n >= 0 with lim_{k -> 1} A(n,k) = A(n,1) = Fibonacci(n).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 5, 7, 4, 1, 1, 0, 1, 8, 31, 13, 5, 1, 1, 0, 1, 13, 255, 121, 21, 6, 1, 1, 0, 1, 21, 8191, 3280, 341, 31, 7, 1, 1, 0, 1, 34, 2097151, 797161, 21845, 781, 43, 8, 1, 1, 0, 1, 55, 17179869184, 5230176601, 22369621, 97656, 1555, 57, 9, 1, 1, 0

Offset: 0

Werner Schulte, May 18 2021

Replacing Fibonacci(n), A000045, with Lucas(n), A000032, you get another square array B(n,k). The terms satisfy the same recurrence equation B(n,k) = (k - 1) * B(n-1,k) * B(n-2,k) + B(n-1,k) + B(n-2,k) for k >= 0 and n > 1 with initial values B(0,k) = k+1 and B(1,k) = 1. Please take account of lim_{k -> 1} (k^Lucas(n) - 1) / (k - 1) = Lucas(n).

			The array A(n,k) for k >= 0 and n >= 0 begins:
n \ k: 0  1           2          3        4     5    6    7  8  9  10  11
=========================================================================
   0 : 0  0           0          0        0     0    0    0  0  0   0   0
   1 : 1  1           1          1        1     1    1    1  1  1   1   1
   2 : 1  1           1          1        1     1    1    1  1  1   1   1
   3 : 1  2           3          4        5     6    7    8  9 10  11  12
   4 : 1  3           7         13       21    31   43   57 73 91 111 133
   5 : 1  5          31        121      341   781 1555 2801
   6 : 1  8         255       3280    21845 97656
   7 : 1 13        8191     797161 22369621
   8 : 1 21     2097151 5230176601
   9 : 1 34 17179869184
  10 : 1 55
  11 : 1 89
  etc.

Cf. A011655 (column k = -1), A057427 (column 0), A000045 (column 1), A063896 (column 2), A000004 (row 0), A000012 (rows 1, 2), A000027 (row 3), A002061 (row 4), A053699 (row 5), A053717 (row 6), A060887 (row 7).

A(n,k) = (k - 1) * A(n-1,k) * A(n-2,k) + A(n-1,k) + A(n-2,k) for k >= 0 and n > 1 with initial values A(0,k) = 0 and A(1,k) = 1.

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A020521 13th cyclotomic polynomial evaluated at powers of 2.

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